METRIC AND TOPO-GEOMETRIC PROPERTIES OF URBAN STREET NETWORKS : some convergences , divergences , and new results
(2007)

The theory of cities, which has grown out of the use of space syntax techniques in urban studies, proposes a curious mathematical duality: that urban space is locally metric but globally topo-geometric. Evidence for local metricity comes from such generic phenomena as grid intensification to reduce mean trip lengths in live centres, the fall of movement from attractors with metric distance, and the commonly observed decay of shopping with metric distance from an intersection. Evidence for global topo-geometry come from the fact that we need to utilise both the geometry and connectedness of the larger scale space network to arrive at configurational measures which optimally approximate movement patterns in the urban network. It might be conjectured that there is some threshold above which human being use some geometrical and topological representation of the urban grid rather than the sense of bodily distance to making movement decisions, but this is unknown. The discarding of metric properties in the large scale urban grid has, however, been controversial. Here we cast a new light on this duality. We show first some phenomena in which metric and topo-geometric measures of urban space converge and diverge, and in doing so clarify the relation between the metric and topo-geometric properties of urban spatial networks. We then show how metric measures can be used to create a new urban phenomenon: the partitioning of the background network of urban space into a network of semi-discrete patches by applying metric universal distance measures at different metric radii, suggesting a natural spatial area-isation of the city at all scales. On this basis we suggest a key clarification of the generic structure of cities: that metric universal distance captures exactly the formally and functionally local patchwork properties of the network, most notably the spatial differentiation of areas, while the top-geometric measures identifying the structure which overcomes locality and links the urban patchwork into a whole at different scales. Introduction: the dual urban network The theory of cities, which has grown out of the use of space syntax techniques in urban studies, proposes that urban street networks have a dual form: a foreground network of linked centres at all scales, and a background network of primarily residential space in which the foreground network is embedded. (Hillier 2001/2) The theory also notes a mathematical duality. On the one hand, measures which express the geometric and topological properties of the network at an extended scale, such as integration and choice measures in axial maps or segment angular maps, are needed to capture structure-function relations such as natural movement patterns (Hillier & Iida 2005). We can call these measures topo-geometric. On the other, at a more localised level, an understanding of structure-function relations often requires an account of metric properties – for example the generic, but usually local, phenomenon of grid intensification to reduce mean trip lengths in live centres (Siksna 1997, Hillier 1999), the fall of movement rates with metric distance from attractors, and the commonly observed decay of shopping with metric distance from an intersection. In terms of understanding structure-function relations. urban space seems to be globally topo-geometric but locally metric. Here we propose to link these two dualities in a more thorough-going way. We show first that the large scale foreground network of space in cities, in spite of the claims of critics (Ratti 2004), really is not metric. On the contrary, the substitution of metric for topo-geometric measures in the analysis, has catastrophic effects on the ability of syntax to account for structure-function relations at this scale. At the same time, topo-geometric measures turn out to capture some interesting metric properties of the larger scale urban network. But we then show that the background network of space really is metric in a much more general sense than has been thought, in that metric measures at different radii can be used to partition the background network of urban space into a patchwork of semi-discrete areas, suggesting a natural metric area-isation of cities at all scales as a function of the placing, shaping and scaling of urban blocks. On this basis we suggest a clarification of the dual structure of cities: that metric ‘universal distance’ (distance from all points to all others – Hillier 1996) measures can capture the spatial differentiation of the background urban network into a patchwork of local areas, while the topo-geometric measures identify the structures which overcomes locality and links the urban patchwork into a whole at different scales. The patchwork theory is in effect a theory of block size and shape, picking up the local distortions in urban space induced by the placing and shaping of physical structures. More generally, we can say that the local-to-global topo-geometric structure reflects the visual, and so non-local effects of placing blocks in space, while the patchwork structure reflects metric and so local effects. The patchwork theory extends and generalises the concept of grid intensification, meaning the reduction of block size to reduce mean distance from all points to all others in a space network. As shown in (Hillier 2000), holding total land coverage and travellable distance in free space constant, a grid in which smaller blocks are placed at the centre and larger blocks at the edge has lower mean distance from all points to all others in the space network than a regular grid, while if larger blocks are placed at the centre and smaller blocks at the edge, then the mean distance from all points to others in the space network is higher than in a regular grid. This follows from the partitioning theory set out in Chapter 8 of Space is the Machine. (Hillier 1996) In general in urban grids, live centres and sub-centres (‘live’ in the sense of having movement dependent uses such as retail and catering) tend to the grid intensified form to maximise the inter-accessibility of the facilities within in the centre, residential areas tend to larger block sizes, reflecting the need to restrain and structure movement in the image of a spatial culture, while the linkages between centres tend to an even larger block size again, an effect of the directional structuring of routes, so that the network of linked centres which dominate the spatial structure of cities tend to oscillate between a relatively large and relative small block size, with the residential background occupying the middle range. This block size pattern is explained more fully in (Hillier 2001/2). In this paper we: – First review the duality of urban space in three ways: geometrically to establish its empirical existence as a key dimension of urban form, functionally to show its implications in terms of movement and land use patterns, and syntactically to show the relations between the two. – We then explore some of the suggestions that have been made about reducing the distance between syntax and more traditional metric approaches, in particular by examining the suggestion of Ratti that we should add metric weightings to the main syntax measures. We show the consequences of these suggestions for any theory which seeks to identify functionally meaningful structures in urban space – We then suggest a general method for showing the metric effect on space of block placing and shaping, both visually and in terms of patterns in scattergrams, by showing theoretical cases – We then apply this method to some cities and show its ability to identify if not natural spatial areas then at least a natural periodicity in city networks through which they tend to a natural spatial area-isation at all scales, reflecting the ways in which we talks about urban areas and regions at different scales. Metric and geometric properties of the grid First, we consider the urban duality geometrically by looking sections of metropolitan Tokyo and London, shown in Figure 1. As shown in (Hillier 2001/2) and later formalised in (Carvalho & Penn 2004), we must first remind ourselves of the fractal nature of urban least line networks: all are made up at all scales, from the local area to the city regions, of a small number of long lines and a large number of short lines. But there is more to be said. Longer and shorter lines form different kinds of geometric patterns. If we look for patterns in the section of Tokyo, the first thing the eye notes are line continuities. What we are seeing in effect is sequence of lines linked at their ends by nearly straight intersections with other lines, forming a visually dominant pattern in the network. But in general, the lines forming these nearly straight continuities as Figueredo calls them (Figueredo 2003) are longer than other nearby lines. This has the effect that if we find a locally longer line it is likely that at either end it will lead to another to which it will be connected by a nearly straight connection, and these lines will in turn be similarly connected. Probabilistically, we can say the longer the line, the more likely it is to end in a nearly straight connection, and taken together these alignments form a network of multi-directional sequences. Intuitively, the value of these in navigating urban grids is obvious, but here, following (Hillier 1999) we are making a structural point. Figure 1: sections of the Tokyo and London street networks What then of the shorter lines ? Again, in spite of a highly variable geometry, we find certain consistencies. First, shorter lines tend to form clusters, so that in the vicinity of each longer line there will be several shorter lines. These localised groups tend to form more grid-like local patterns, with lines either passing through each other, or ending on other lines, at near right angles. We can say then that the shorter the line, the more likely it is to